Show this is a projection on a vector space

JaysFan31

Homework Statement

Let V=Mn(F) be the space of all nxn matrices over F; define TA=(1/2)(A+transpose(A)) for A in V.
Verify that T is not only a linear operator on V, but is also a projection.

Homework Equations

A is a projection when A squared=A.

The Attempt at a Solution

I don't see how this works since clearly (1/2)(A+transpose(A)) squared does not equal (1/2)(A+transpose(A)) for all matrices.

What am I doing wrong?

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AKG
$$\left [\frac{1}{2}(A + A^t)\right ]^2 = \frac{1}{2}(A + A^t)$$
$$\frac{1}{2}\left [\left (\frac{1}{2}(A + A^t)\right ) + \left (\frac{1}{2}(A + A^t)\right )^t\right ] = \frac{1}{2}(A + A^t)$$